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      "metadata": {
        "id": "rx7xM35VlNk0",
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      "source": [
        "# Support vector machine classifier with $\\ell_1$-regularization"
      ]
    },
    {
      "metadata": {
        "id": "UBeCbr-dlNk2",
        "colab_type": "text"
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      "source": [
        "In this example we use CVXPY to train a SVM classifier with $\\ell_1$-regularization.\n",
        "We are given data $(x_i,y_i)$, $i=1,\\ldots, m$. The $x_i \\in {\\bf R}^n$ are feature vectors, while the $y_i \\in \\{\\pm 1\\}$ are associated boolean outcomes.\n",
        "Our goal is to construct a good linear classifier $\\hat y = {\\rm sign}(\\beta^T x - v)$.\n",
        "We find the parameters $\\beta,v$ by minimizing the (convex) function\n",
        "\n",
        "$$\n",
        "f(\\beta,v) = (1/m) \\sum_i \\left(1 - y_i ( \\beta^T x_i-v) \\right)_+ + \\lambda\n",
        "\\| \\beta\\|_1\n",
        "$$\n",
        "\n",
        "The first term is the average hinge loss. The second term shrinks the coefficients in $\\beta$ and encourages sparsity.\n",
        "The scalar $\\lambda \\geq 0$ is a (regularization) parameter.\n",
        "Minimizing $f(\\beta,v)$  simultaneously selects features and fits the classifier."
      ]
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      "metadata": {
        "id": "7PlzylpWlNk3",
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      "source": [
        "### Example\n",
        "\n",
        "In the following code we generate data with $n=20$ features by randomly choosing $x_i$ and a sparse $\\beta_{\\mathrm{true}} \\in {\\bf R}^n$.\n",
        "We then set $y_i = {\\rm sign}(\\beta_{\\mathrm{true}}^T x_i -v_{\\mathrm{true}} - z_i)$, where the $z_i$ are i.i.d. normal random variables.\n",
        "We divide the data into training and test sets with $m=1000$ examples each."
      ]
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    {
      "metadata": {
        "id": "DxXlfuzLlNk3",
        "colab_type": "code",
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      "cell_type": "code",
      "source": [
        "# Generate data for SVM classifier with L1 regularization.\n",
        "from __future__ import division\n",
        "import numpy as np\n",
        "np.random.seed(1)\n",
        "n = 20\n",
        "m = 1000\n",
        "TEST = m\n",
        "DENSITY = 0.2\n",
        "beta_true = np.random.randn(n,1)\n",
        "idxs = np.random.choice(range(n), int((1-DENSITY)*n), replace=False)\n",
        "for idx in idxs:\n",
        "    beta_true[idx] = 0\n",
        "offset = 0\n",
        "sigma = 45\n",
        "X = np.random.normal(0, 5, size=(m,n))\n",
        "Y = np.sign(X.dot(beta_true) + offset + np.random.normal(0,sigma,size=(m,1)))\n",
        "X_test = np.random.normal(0, 5, size=(TEST,n))\n",
        "Y_test = np.sign(X_test.dot(beta_true) + offset + np.random.normal(0,sigma,size=(TEST,1)))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "dezeHwQDlNk6",
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      "cell_type": "markdown",
      "source": [
        "We next formulate the optimization problem using CVXPY."
      ]
    },
    {
      "metadata": {
        "id": "gp278JAclNk7",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Form SVM with L1 regularization problem.\n",
        "import cvxpy as cp\n",
        "beta = cp.Variable((n,1))\n",
        "v = cp.Variable()\n",
        "loss = cp.sum(cp.pos(1 - cp.multiply(Y, X*beta - v)))\n",
        "reg = cp.norm(beta, 1)\n",
        "lambd = cp.Parameter(nonneg=True)\n",
        "prob = cp.Problem(cp.Minimize(loss/m + lambd*reg))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "v7clejmAlNk9",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "We solve the optimization problem for a range of $\\lambda$ to compute a trade-off curve.\n",
        "We then plot the train and test error over the trade-off curve.\n",
        "A reasonable choice of $\\lambda$ is the value that minimizes the test error."
      ]
    },
    {
      "metadata": {
        "id": "URUg5liflNk-",
        "colab_type": "code",
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      },
      "cell_type": "code",
      "source": [
        "# Compute a trade-off curve and record train and test error.\n",
        "TRIALS = 100\n",
        "train_error = np.zeros(TRIALS)\n",
        "test_error = np.zeros(TRIALS)\n",
        "lambda_vals = np.logspace(-2, 0, TRIALS)\n",
        "beta_vals = []\n",
        "for i in range(TRIALS):\n",
        "    lambd.value = lambda_vals[i]\n",
        "    prob.solve()\n",
        "    train_error[i] = (np.sign(X.dot(beta_true) + offset) != np.sign(X.dot(beta.value) - v.value)).sum()/m\n",
        "    test_error[i] = (np.sign(X_test.dot(beta_true) + offset) != np.sign(X_test.dot(beta.value) - v.value)).sum()/TEST\n",
        "    beta_vals.append(beta.value)"
      ],
      "execution_count": 0,
      "outputs": []
    },
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        "outputId": "2d9bd5f7-d7db-4fc6-87fc-b92fcc2ff8d8"
      },
      "cell_type": "code",
      "source": [
        "# Plot the train and test error over the trade-off curve.\n",
        "import matplotlib.pyplot as plt\n",
        "%matplotlib inline\n",
        "%config InlineBackend.figure_format = 'svg'\n",
        "\n",
        "plt.plot(lambda_vals, train_error, label=\"Train error\")\n",
        "plt.plot(lambda_vals, test_error, label=\"Test error\")\n",
        "plt.xscale('log')\n",
        "plt.legend(loc='upper left')\n",
        "plt.xlabel(r\"$\\lambda$\", fontsize=16)\n",
        "plt.show()"
      ],
      "execution_count": 8,
      "outputs": [
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        "id": "nLca9p-ulNlF",
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      "cell_type": "markdown",
      "source": [
        "We also plot the regularization path, or the $\\beta_i$ versus $\\lambda$. Notice that the $\\beta_i$ do not necessarily decrease monotonically as $\\lambda$ increases.\n",
        "4 features remain non-zero longer for larger $\\lambda$ than the rest, which suggests that these features are the most important. In fact $\\beta_{\\mathrm{true}}$ had 4 non-zero values."
      ]
    },
    {
      "metadata": {
        "id": "9FfvOlADlNlF",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 383
        },
        "outputId": "b904b3a7-6c5d-4511-8095-06c84ff9acd2"
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      "cell_type": "code",
      "source": [
        "# Plot the regularization path for beta.\n",
        "for i in range(n):\n",
        "    plt.plot(lambda_vals, [wi[i,0] for wi in beta_vals])\n",
        "plt.xlabel(r\"$\\lambda$\", fontsize=16)\n",
        "plt.xscale(\"log\")"
      ],
      "execution_count": 9,
      "outputs": [
        {
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